Recall that [1]} for a split inclusion \((\mathcal {N}_{\mathrm {c}}\subset \mathcal {M}_{\mathrm {c}}, \Omega _{\mathrm {c}})\)  equipped with a cyclic separating
vector \(\Omega _{\mathrm {c}}\)  for \(\mathcal {M}_{\mathrm {c}}, \mathcal {N}_{\mathrm {c}}\)  and \(\mathcal {M}_{\mathrm {c}}\cap \mathcal {N}_{\mathrm {c}}\)  (where \(\mathcal {N}_{\mathrm {c}}\)  will be \({\hbox{\rm Ad\,}}T_{\mathrm {c}}(a)(\mathcal {M}_{\mathrm {c}})\)  for some \(a \in W_{\mathrm {R}}\) ),
there is a canonical type I factor \(\mathcal {R}_{\mathrm {c}}\) 
such that \(\mathcal {N}_{\mathrm {c}}\subset \mathcal {R}_{\mathrm {c}}\subset \mathcal {M}_{\mathrm {c}}\) . Moreover, \(\mathcal {R}_{\mathrm {c}}\)  is given by the formula
\(\mathcal {R}_{\mathrm {c}}= \mathcal {N}_{\mathrm {c}}\vee J\mathcal {N}_{\mathrm {c}}J = \mathcal {M}_{\mathrm {c}}\cap J\mathcal {M}_{\mathrm {c}}J\) , where \(J\)  is the modular conjugation for \(\mathcal {M}_{\mathrm {c}}\cap \mathcal {N}_{\mathrm {c}}\)  with
respect to \(\Omega _{\mathrm {c}}\) .
If \(V_{\mathrm {c}}\)  is a unitary operator which preserves \(\Omega _{\mathrm {c}}\) 
and \({\hbox{\rm Ad\,}}V_{\mathrm {c}}\)  preserves both \(\mathcal {N}_{\mathrm {c}}\)  and \(\mathcal {M}_{\mathrm {c}}\) , then \(\mathcal {R}_{\mathrm {c}}\)  is preserved under \({\hbox{\rm Ad\,}}V_{\mathrm {c}}\)  as well.
